91Ó°ÊÓ

The phrase 'initial amount' refers to the quantity of a substance when the observation starts, often at time zero. This would be represented as the value of A(t) when t = 0 in our exponential decay formula.
For the given problem, the formula is:
\[ A(t) = 35 \, e^{-0.029t} \]
To find the initial amount, we substitute \(t = 0\) into the equation:
\[ A(0) = 35 \, e^{-0.029 \times 0} = 35 \]
This calculation shows us that the initial amount of thorium-234 is 35 micrograms. This initial value sets the stage for studying how the substance decreases over time through exponential decay. Understanding the initial amount is essential as it serves as a reference for future measurements and calculations.

The 'half-life' of a substance is the time it takes for half of the initial amount to decay. This concept is crucial for understanding how quickly a substance decreases in quantity. In our thorium-234 problem, we need to find when the amount becomes half of the initial 35 micrograms.
This involves solving for the time \(t\) when \( A(t) = \frac{35}{2} = 17.5 \). Using the decay formula again:
\[ 17.5 = 35 \, e^{-0.029t} \] \[ \frac{1}{2} = e^{-0.029t} \]
Applying the natural logarithm to both sides gives:
\[ \text{ln}\frac{1}{2} = \text{ln}(e^{-0.029t}) \] Simplifying, we get:
\[ -0.693 = -0.029t \] Which further solves to:
\[ t = \frac{-0.693}{-0.029} \]
Hence, the half-life of thorium-234 is approximately 23.9 days. Knowing the half-life helps predict how long it will take for any given quantity to halve, a key aspect in fields like medicine, archaeology, and nuclear physics.

The 'natural logarithm', denoted as ln, is a logarithm to the base e (where e is approximately equal to 2.71828). The natural logarithm is vital for solving exponential equations, such as those involving decay processes.
In our problem, we use the natural logarithm to solve for time when given a certain remaining amount. For example, to find when there is just 1 microgram left:
\[ 1 = 35 \, e^{-0.029t} \]
Divide both sides by 35:
\[ \frac{1}{35} = e^{-0.029t} \]
Taking the natural logarithm of both sides lets us use the property \( \text{ln}(e^x) = x \):
\[ \text{ln}\frac{1}{35} = \text{ln}(e^{-0.029t}) \]
Which simplifies to:
\[ -3.555 = -0.029t \]
Solving for t gives:
\[ t = \frac{-3.555}{-0.029} \]
Thus, t ≈ 122.59 days. The natural logarithm transforms the exponential relationship into a linear one, making it easier to solve for variables like time.

The 'decay constant' (denoted as λ or in our case as -0.029), is a value that characterizes the rate at which a substance decays exponentially. The larger the decay constant, the faster the substance decreases.
Here, the decay constant \( -0.029 \) indicates how rapidly thorium-234 is decaying over time. It's found in the exponent of our decay formula:
\[ A(t) = 35 \, e^{-0.029t} \]
The decay constant allows us to compute other essential parameters such as half-life and the remaining amount at any time t.
For example, to find the half-life, knowing the decay constant enables us to set up the equation:
\[ \frac{1}{2} = e^{-0.029t} \]
And then solve for t. This constant forms the backbone of all computations relating to exponential decay in any scenario where the rate of decrease is proportional to the current amount present.